Unifying Practical Uncertainty Representations: II. Clouds

TL;DR

This paper explores the relationships between clouds and other uncertainty representations, showing that generalized p-boxes are clouds with comonotonic distributions, but clouds cannot always be represented by random sets or convex capacities.

Contribution

It establishes the connection between clouds and generalized p-boxes, and clarifies the limitations of representing clouds with random sets or convex capacities.

Findings

Generalized p-boxes are equivalent to clouds with comonotonic distributions.

Clouds cannot always be represented by random sets.

Clouds are not always representable by 2-monotone capacities.

Abstract

There exist many simple tools for jointly capturing variability and incomplete information by means of uncertainty representations. Among them are random sets, possibility distributions, probability intervals, and the more recent Ferson's p-boxes and Neumaier's clouds, both defined by pairs of possibility distributions. In the companion paper, we have extensively studied a generalized form of p-box and situated it with respect to other models . This paper focuses on the links between clouds and other representations. Generalized p-boxes are shown to be clouds with comonotonic distributions. In general, clouds cannot always be represented by random sets, in fact not even by 2-monotone (convex) capacities.